Lagrange's Formula

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Theorem

Let:

$\mathbf a = \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix}$, $\mathbf b = \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix}$, $\mathbf c = \begin{bmatrix} c_x \\ c_y \\ c_z \end{bmatrix}$

be vectors in a vector space of $3$ dimensions.


Then:

$\mathbf a \times \left({\mathbf b \times \mathbf c}\right) = \left({\mathbf{a \cdot c} }\right) \mathbf b - \left({\mathbf{a \cdot b} }\right) \mathbf c$


Corollary

$\left({\mathbf a \times \mathbf b}\right) \times \mathbf c = \left({\mathbf{a \cdot c} }\right) \mathbf b - \left({\mathbf{b \cdot c} }\right) \mathbf a$


Proof

\(\displaystyle \mathbf{b \times c}\) \(=\) \(\displaystyle \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix} \times \begin{bmatrix} c_x \\ c_y \\ c_z \end{bmatrix}\)
\(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} b_y c_z - b_z c_y \\ b_z c_x - b_x c_z \\ b_x c_y - b_y c_x \end{bmatrix}\) Definition of Vector Cross Product
\(\displaystyle \mathbf a \times \left({\mathbf{b \times c} }\right)\) \(=\) \(\displaystyle \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix} \times \begin{bmatrix} b_y c_z - b_z c_y \\ b_z c_x - b_x c_z \\ b_x c_y - b_y c_x \end{bmatrix}\)
\(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} a_y b_x c_y - a_y b_y c_x - a_z b_z c_x + a_z b_x c_z \\ a_z b_y c_z - a_z b_z c_y - a_x b_x c_y + a_x b_y c_x \\ a_x b_z c_x - a_x b_x c_z - a_y b_y c_z + a_y b_z c_y \end{bmatrix}\) Definition of Vector Cross Product
\(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} a_y b_x c_y - a_y b_y c_x - a_z b_z c_x + a_z b_x c_z + a_x b_x c_x - a_x b_x c_x \\ a_z b_y c_z - a_z b_z c_y - a_x b_x c_y + a_x b_y c_x + a_y b_y c_y - a_y b_y c_y \\ a_x b_z c_x - a_x b_x c_z - a_y b_y c_z + a_y b_z c_y + a_z b_z c_z - a_z b_z c_z \end{bmatrix}\) adding $0 = a_i b_i c_i - a_i b_i c_i$ to each entry
\(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} b_x \left({a_y c_y + a_z c_z + a_x c_x}\right) - c_x\left({a_y b_y + a_z b_z + a_x b_x}\right) \\ b_y\left({a_z c_z + a_x c_x + a_y c_y}\right) - c_y\left({a_z b_z + a_x b_x + a_y c_y}\right)\\ b_z\left({a_x c_x + a_y c_y + a_z c_z}\right) - c_z\left({a_x b_x + a_y b_y + a_z c_z}\right) \end{bmatrix}\)
\(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} b_x \left({\mathbf{a \cdot c} }\right) - c_x \left({\mathbf {a \cdot b} }\right) \\ b_y \left({\mathbf{a \cdot c} }\right) - c_y \left({\mathbf {a \cdot b} }\right) \\ b_z \left({\mathbf{a \cdot c} }\right) - c_z \left({\mathbf {a \cdot b} }\right) \end{bmatrix}\) Definition of Dot Product
\(\displaystyle \) \(=\) \(\displaystyle \left({\mathbf{a \cdot c} }\right) \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix} - \left({\mathbf{a \cdot b} }\right) \begin{bmatrix} c_x \\ c_y \\ c_z \end{bmatrix}\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\mathbf{a \cdot c} }\right) \mathbf b - \left({\mathbf{a \cdot b} }\right) \mathbf c\)

$\blacksquare$


Source of Name

This entry was named for Joseph Louis Lagrange.


Sources